Physical Review A

Third-order perturbation theory for van der Waals interaction coefficients

Li-Yan Tang, Zong-Chao Yan, Ting-Yun Shi, Jim Mitroy

Published Nov 4, 2011• 5 citations

Abstract

The third-order expression for the dispersion interaction between two atoms is written as a sum over lists of transition matrix elements. Particular attention is given to the ${C}_{9}/{R}^{9}$ interaction which occurs in the homonuclear case when one atom is in an $S$ state and the other is in a $P$ state. Numerical values of the ${C}_{9}$ coefficient are given for the homonuclear alkali-metal dimers. The size of the ${C}_{9}$:${C}_{3}$ dispersion coefficient ratio increases for the heavier alkali-metal atoms. The ${C}_{11}$ and ${C}_{13}$ coefficients between two helium atoms and lithium atoms in their ground states are also given.

DOI: 10.1103/physreva.84.052502

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Physical Review A

Third-order perturbation theory for van der Waals interaction coefficients

Li-Yan Tang, Zong-Chao Yan, Ting-Yun Shi, Jim Mitroy

Published Nov 4, 2011• 5 citations

Publisher

Abstract

The third-order expression for the dispersion interaction between two atoms is written as a sum over lists of transition matrix elements. Particular attention is given to the

{C}_{9}/{R}^{9}

interaction which occurs in the homonuclear case when one atom is in an

S

state and the other is in a

P

state. Numerical values of the

{C}_{9}

coefficient are given for the homonuclear alkali-metal dimers. The size of the

{C}_{9}

{C}_{3}

dispersion coefficient ratio increases for the heavier alkali-metal atoms. The

{C}_{11}

and

{C}_{13}

coefficients between two helium atoms and lithium atoms in their ground states are also given.

DOI: 10.1103/physreva.84.052502

Related papers

Higher-order Cn dispersion coefficients for the alkali-metal atoms

J. Mitroy, M. W. J. Bromley

The van der Waals coefficients, from ${C}_{11}$ through to ${C}_{16}$ resulting from second-, third-, and fourth-order perturbation theory are estimated for the alkali-metal (Li, Na, K, and Rb) atoms. The dispersion coefficients are also computed for all possible combinations of the alkali-metal atoms and hydrogen. The parameters are determined from sum rules after diagonalizing a semiempirical fixed core Hamiltonian in a large basis. Comparisons of the radial dependence of the ${C}_{n}∕{r}^{n}$ potentials give guidance as to the radial regions in which the various higher-order terms can be neglected. It is seen that including terms up to ${C}_{10}∕{r}^{10}$ results in a dispersion interaction that is accurate to better than 1% whenever the inter-nuclear spacing is larger than $20{a}_{0}$ . This level of accuracy is mainly achieved due to the fortuitous cancellation between the repulsive $({C}_{11},{C}_{13},{C}_{15})$ and attractive $({C}_{12},{C}_{14},{C}_{16})$ dispersion forces.The van der Waals coefficients, from ${C}_{11}$ through to ${C}_{16}$ resulting from second-, third-, and fourth-order perturbation theory are estimated for the alkali-metal (Li, Na, K, and Rb) atoms. The dispersion coefficients are also computed for all possible combinations of the alkali-metal atoms and hydrogen. The parameters are determined from sum rules after diagonalizing a semiempirical fixed core Hamiltonian in a large basis. Comparisons of the radial dependence of the ${C}_{n}∕{r}^{n}$ potentials give guidance as to the radial regions in which the various higher-order terms can be neglected. It is seen that including terms up to ${C}_{10}∕{r}^{10}$ results in a dispersion interaction that is accurate to better than 1% whenever the inter-nuclear spacing is larger than $20{a}_{0}$. This level of accuracy is mainly achieved due to the fortuitous cancellation between the repulsive $({C}_{11},{C}_{13},{C}_{15})$ and attractive $({C}_{12},{C}_{14},{C}_{16})$ dispersion forces.

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Long-range interactions between excited helium and alkali-metal atoms

J. Y. Zhang, Li-Yan Tang, T.-Y. Shi +2 more

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Semiempirical calculation of van der Waals coefficients for alkali-metal and alkaline-earth-metal atoms

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Dispersion coefficients for the nP - nP asymptote of homonuclear alkali-metal dimers

M. Marinescu

The derivation of the dispersion coefficients for various symmetries of the lower $\mathrm{nP}$ - $\mathrm{nP}$ asymptote of the homonuclear alkali-metal dimers is presented. Using the perturbation theory the dispersion coefficients ${C}_{5}$ are obtained in the first order and the ${C}_{6}$ and ${C}_{8}$ coefficients are obtained in the second order in terms of double sums involving atomic radial matrix elements. The sums are separated in individual contributions of each atom by using a complex integral representation. The atomic matrix elements involving Green's functions are evaluated using the Dalgarno-Lewis method. Numerical results for the ${C}_{5}$ , ${C}_{6}$ , and ${C}_{8}$ dispersion coefficients for Li, Na, K, Rb, and Cs dimers are presented.The derivation of the dispersion coefficients for various symmetries of the lower $\mathrm{nP}$-$\mathrm{nP}$ asymptote of the homonuclear alkali-metal dimers is presented. Using the perturbation theory the dispersion coefficients ${C}_{5}$ are obtained in the first order and the ${C}_{6}$ and ${C}_{8}$ coefficients are obtained in the second order in terms of double sums involving atomic radial matrix elements. The sums are separated in individual contributions of each atom by using a complex integral representation. The atomic matrix elements involving Green's functions are evaluated using the Dalgarno-Lewis method. Numerical results for the ${C}_{5}$, ${C}_{6}$, and ${C}_{8}$ dispersion coefficients for Li, Na, K, Rb, and Cs dimers are presented.

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Dispersion coefficients for alkali-metal dimers

M. Marinescu, H. R. Sadeghpour, A. Dalgarno

Knowledge of the long-range interaction between atoms and molecules is of fundamental importance for low-energy and low-temperature collisions. The electronic interaction between the charge distributions of two ground-state alkali-metal atoms can be expanded in inverse powers of R, the internuclear distance. The coefficients ${\mathit{C}}_{6}$ , ${\mathit{C}}_{8}$ , and ${\mathit{C}}_{10}$ of, respectively, the ${\mathit{R}}^{\mathrm{\ensuremath{-}}6}$, ${\mathit{R}}^{\mathrm{\ensuremath{-}}8}$, and ${\mathit{R}}^{\mathrm{\ensuremath{-}}10}$ terms are calculated by integrating the products of the dynamic electric multipole polarizabilities of the individual atoms at imaginary frequencies, which are in turn obtained by solving two coupled inhomogeneous differential equations. Precise one-electron model potentials are developed to represent the motion of the valence electron in the field of the closed alkali-metal positive-ion core. The numerical results for the static multipole polarizabilities for the alkali-metal atoms and the coefficients ${\mathit{C}}_{6}$ , ${\mathit{C}}_{8}$ , and ${\mathit{C}}_{10}$ for homonuclear and heteronuclear alkali-metal diatoms are compared with other calculations.Knowledge of the long-range interaction between atoms and molecules is of fundamental importance for low-energy and low-temperature collisions. The electronic interaction between the charge distributions of two ground-state alkali-metal atoms can be expanded in inverse powers of R, the internuclear distance. The coefficients ${\mathit{C}}_{6}$, ${\mathit{C}}_{8}$, and ${\mathit{C}}_{10}$ of, respectively, the ${\mathit{R}}^{\mathrm{\ensuremath{-}}6}$, ${\mathit{R}}^{\mathrm{\ensuremath{-}}8}$, and ${\mathit{R}}^{\mathrm{\ensuremath{-}}10}$ terms are calculated by integrating the products of the dynamic electric multipole polarizabilities of the individual atoms at imaginary frequencies, which are in turn obtained by solving two coupled inhomogeneous differential equations. Precise one-electron model potentials are developed to represent the motion of the valence electron in the field of the closed alkali-metal positive-ion core. The numerical results for the static multipole polarizabilities for the alkali-metal atoms and the coefficients ${\mathit{C}}_{6}$, ${\mathit{C}}_{8}$, and ${\mathit{C}}_{10}$ for homonuclear and heteronuclear alkali-metal diatoms are compared with other calculations.

Physical Review AFeb 1, 1994377 citations

van der Waals interaction in excited alkali-metal dimers

Mladen Movre, Robert Beuc

We have studied the long-range interaction potentials of the heteronuclear alkali-metal-atom dimers in the case when one of the atoms is in the first excited state ( ${n}_{A}$ P) and the other is in the ground state ( ${n}_{B}$ S). The presentation of the general expressions for the interaction energies in the Hund's case (c) is followed by the calculation of the corresponding ${C}_{6}$ constants for all heteronuclear alkali-metal-atom pairs. In the case of heavier perturbers (Rb and Cs) the ${C}_{6}$ constants are found to be significantly influenced by the perturber's fine structure. The van der Waals interaction in the K-Rb system is exceptionally strong resulting in ${C}_{6}$ constants considerably larger than in all other alkali-metal-atom heteronuclear systems. The intermediate-coupling region (ICR) is defined as the region of internuclear separation R where the interaction energy is of the order of the atomic spin-orbit splitting, while the exchange energy is still negligible. Analysis of the general form of the energy matrices in the ICR shows that the homonuclear alkali-metal-atom dimers can be treated as a special case. In the large-R limit, the ICR results reduce to those obtained in Hund's case (c). As an example, we have calculated the ICR interaction potentials of the K-Rb system, taking into account only the leading dipole-dipole interaction. Finally we have discussed the construction of the absorption coefficient in the wings of the resonance lines, both in the impact and in the quasistatic approximation.We have studied the long-range interaction potentials of the heteronuclear alkali-metal-atom dimers in the case when one of the atoms is in the first excited state (${n}_{A}$P) and the other is in the ground state (${n}_{B}$S). The presentation of the general expressions for the interaction energies in the Hund's case (c) is followed by the calculation of the corresponding ${C}_{6}$ constants for all heteronuclear alkali-metal-atom pairs. In the case of heavier perturbers (Rb and Cs) the ${C}_{6}$ constants are found to be significantly influenced by the perturber's fine structure. The van der Waals interaction in the K-Rb system is exceptionally strong resulting in ${C}_{6}$ constants considerably larger than in all other alkali-metal-atom heteronuclear systems. The intermediate-coupling region (ICR) is defined as the region of internuclear separation R where the interaction energy is of the order of the atomic spin-orbit splitting, while the exchange energy is still negligible. Analysis of the general form of the energy matrices in the ICR shows that the homonuclear alkali-metal-atom dimers can be treated as a special case. In the large-R limit, the ICR results reduce to those obtained in Hund's case (c). As an example, we have calculated the ICR interaction potentials of the K-Rb system, taking into account only the leading dipole-dipole interaction. Finally we have discussed the construction of the absorption coefficient in the wings of the resonance lines, both in the impact and in the quasistatic approximation.

Physical Review AMay 1, 198530 citations

The third-order expression for the dispersion interaction between two atoms is written as a sum over lists of transition matrix elements. Particular attention is given to the

{C}_{9}/{R}^{9}

interaction which occurs in the homonuclear case when one atom is in an

S

state and the other is in a

P

state. Numerical values of the

{C}_{9}

coefficient are given for the homonuclear alkali-metal dimers. The size of the

{C}_{9}

{C}_{3}

dispersion coefficient ratio increases for the heavier alkali-metal atoms. The

{C}_{11}

and

{C}_{13}

coefficients between two helium atoms and lithium atoms in their ground states are also given.

The van der Waals coefficients, from

{C}_{11}

through to

{C}_{16}

resulting from second-, third-, and fourth-order perturbation theory are estimated for the alkali-metal (Li, Na, K, and Rb) atoms. The dispersion coefficients are also computed for all possible combinations of the alkali-metal atoms and hydrogen. The parameters are determined from sum rules after diagonalizing a semiempirical fixed core Hamiltonian in a large basis. Comparisons of the radial dependence of the

{C}_{n}∕{r}^{n}

potentials give guidance as to the radial regions in which the various higher-order terms can be neglected. It is seen that including terms up to

{C}_{10}∕{r}^{10}

results in a dispersion interaction that is accurate to better than 1% whenever the inter-nuclear spacing is larger than

20{a}_{0}

. This level of accuracy is mainly achieved due to the fortuitous cancellation between the repulsive

({C}_{11},{C}_{13},{C}_{15})

and attractive

({C}_{12},{C}_{14},{C}_{16})

dispersion forces.

The dispersion coefficients for the long-range interaction of the first four excited states of He, i.e., He(2

{}^{1,3}\phantom{\rule{-0.16em}{0ex}}

S

) and He(2

{}^{1,3}

P

), with the low-lying states of the alkali-metal atoms Li, Na, K, and Rb are calculated by summing over the reduced matrix elements of the multipole transition operators. For the interaction between He and Li the uncertainty of the calculations is 0.1--0.5

%

. For interactions with other alkali-metal atoms the uncertainty is 1--3

%

in the coefficient

{C}_{5}

, 1--5

%

in the coefficient

{C}_{6}

, and 1--10

%

in the coefficients

{C}_{8}

and

{C}_{10}

. The dispersion coefficients

{C}_{n}

for the interaction of He(2

{}^{1,3}\phantom{\rule{-0.16em}{0ex}}

S

) and He(2

{}^{1,3}

P

) with the ground-state alkali-metal atoms and for the interaction of He(2

{}^{1,3}\phantom{\rule{-0.16em}{0ex}}

S

) with the alkali-metal atoms in their first

{}^{2}

P

states are presented in this Brief Report. The coefficients for other pairs of atomic states are listed in the Supplemental Material.

The van der Waals coefficients,

{C}_{6},{C}_{8},

and

{C}_{10}

for the alkali-metal (Li, Na, K, and Rb) and alkaline-earth-metal (Be, Mg, Ca, and Sr) atoms are estimated by a combination of ab initio and semiempirical methods. Polarizabilities and atom-wall coefficients are given as a diagnostic check, and the lowest order nonadiabatic dispersion coefficient,

{D}_{8}

and the three-body coefficient,

{C}_{9}

are also presented. The dispersion coefficients are in agreement with the available relativistic many-body perturbation theory calculations. The contribution from the core was included by using constrained sum rules involving the core polarizability and Hartree-Fock expectation values to estimate the f-value distribution.

The derivation of the dispersion coefficients for various symmetries of the lower

\mathrm{nP}

\mathrm{nP}

asymptote of the homonuclear alkali-metal dimers is presented. Using the perturbation theory the dispersion coefficients

{C}_{5}

are obtained in the first order and the

{C}_{6}

and

{C}_{8}

coefficients are obtained in the second order in terms of double sums involving atomic radial matrix elements. The sums are separated in individual contributions of each atom by using a complex integral representation. The atomic matrix elements involving Green's functions are evaluated using the Dalgarno-Lewis method. Numerical results for the

{C}_{5}

{C}_{6}

, and

{C}_{8}

dispersion coefficients for Li, Na, K, Rb, and Cs dimers are presented.

{\mathit{C}}_{6}

{\mathit{C}}_{8}

, and

{\mathit{C}}_{10}

of, respectively, the ${\mathit{R}}^{\mathrm{\ensuremath{-}}6}$, ${\mathit{R}}^{\mathrm{\ensuremath{-}}8}$, and ${\mathit{R}}^{\mathrm{\ensuremath{-}}10}$ terms are calculated by integrating the products of the dynamic electric multipole polarizabilities of the individual atoms at imaginary frequencies, which are in turn obtained by solving two coupled inhomogeneous differential equations. Precise one-electron model potentials are developed to represent the motion of the valence electron in the field of the closed alkali-metal positive-ion core. The numerical results for the static multipole polarizabilities for the alkali-metal atoms and the coefficients

{\mathit{C}}_{6}

{\mathit{C}}_{8}

, and

{\mathit{C}}_{10}

for homonuclear and heteronuclear alkali-metal diatoms are compared with other calculations.

We have studied the long-range interaction potentials of the heteronuclear alkali-metal-atom dimers in the case when one of the atoms is in the first excited state (

{n}_{A}

P) and the other is in the ground state (

{n}_{B}

S). The presentation of the general expressions for the interaction energies in the Hund's case (c) is followed by the calculation of the corresponding

{C}_{6}

constants for all heteronuclear alkali-metal-atom pairs. In the case of heavier perturbers (Rb and Cs) the

{C}_{6}

constants are found to be significantly influenced by the perturber's fine structure. The van der Waals interaction in the K-Rb system is exceptionally strong resulting in

{C}_{6}

constants considerably larger than in all other alkali-metal-atom heteronuclear systems. The intermediate-coupling region (ICR) is defined as the region of internuclear separation R where the interaction energy is of the order of the atomic spin-orbit splitting, while the exchange energy is still negligible. Analysis of the general form of the energy matrices in the ICR shows that the homonuclear alkali-metal-atom dimers can be treated as a special case. In the large-R limit, the ICR results reduce to those obtained in Hund's case (c). As an example, we have calculated the ICR interaction potentials of the K-Rb system, taking into account only the leading dipole-dipole interaction. Finally we have discussed the construction of the absorption coefficient in the wings of the resonance lines, both in the impact and in the quasistatic approximation.